4 1

3_1

5_1

Visit 4 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 4 1's page at Knotilus! Visit 4 1's page at the original Knot Atlas!

4_1 is also known as "the Figure Eight knot", as some people think it looks like a figure `8' in one of its common projections. See e.g. [1] . For two 4_1 knots along a closed loop, see 10_59, 10_60, K12a975, and K12a991.

Square depiction	Alternate square depiction	3D depiction	In "figure 8" form
A Neli-Kolam with 3x2 dot array[1]	In curved symmetrical form	Quasi-Celtic depiction	Symmetrical from parametric equation
Thurston's Trick [2]	Cylindrical depiction

Non-prime (compound) versions

• 

  Pseudo-Celtic ornamental knot pattern, with three figure-8 knots along a closed triangular loop.

• 

  (alternative)

• 

  Coat of arms surrounded by figure-8 knots

Knot presentations

Planar diagram presentation	X4251 X8615 X6374 X2738
Gauss code	1, -4, 3, -1, 2, -3, 4, -2
Dowker-Thistlethwaite code	4 6 8 2
Conway Notation	[22]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-3] Hyperbolic Volume 2.02988 A-Polynomial See Data:4 1/A-polynomial

[edit Notes for 4 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(4,1)) }[/math] Rasmussen s-Invariant 0

[edit Notes for 4 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t- t^{-1} +3 }[/math]
Conway polynomial	[math]\displaystyle{ 1-z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 5, 0 }
Jones polynomial	[math]\displaystyle{ q^2+ q^{-2} -q- q^{-1} +1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^2+ a^{-2} -z^2-1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1 }[/math]
The A2 invariant	[math]\displaystyle{ q^8+q^6-1+ q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 4_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^5+ q^{-5} }[/math]
2	[math]\displaystyle{ q^{14}-q^{10}+q^2+1+ q^{-2} - q^{-10} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{27}-q^{23}-q^{21}+q^{17}+q^{11}+q^9+ q^{-9} + q^{-11} + q^{-17} - q^{-21} - q^{-23} + q^{-27} }[/math]
4	[math]\displaystyle{ q^{44}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}+q^{24}+q^{22}-q^{18}-q^{16}+q^4+q^2+1+ q^{-2} + q^{-4} - q^{-16} - q^{-18} + q^{-22} + q^{-24} - q^{-26} + q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} - q^{-40} + q^{-44} }[/math]
5	[math]\displaystyle{ q^{65}-q^{61}-q^{59}-q^{57}+q^{53}+2 q^{51}+q^{49}-q^{45}-q^{43}+q^{39}+q^{37}-q^{35}-2 q^{33}-q^{31}+q^{27}+q^{25}+q^{17}+q^{15}+q^{13}+ q^{-13} + q^{-15} + q^{-17} + q^{-25} + q^{-27} - q^{-31} -2 q^{-33} - q^{-35} + q^{-37} + q^{-39} - q^{-43} - q^{-45} + q^{-49} +2 q^{-51} + q^{-53} - q^{-57} - q^{-59} - q^{-61} + q^{-65} }[/math]
6	[math]\displaystyle{ q^{90}-q^{86}-q^{84}-q^{82}+2 q^{76}+2 q^{74}+q^{72}-q^{68}-2 q^{66}-2 q^{64}+q^{62}+q^{60}+q^{58}-q^{54}-2 q^{52}-2 q^{50}+q^{48}+2 q^{46}+2 q^{44}+q^{42}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}-q^{26}-q^{24}-q^{22}+q^6+q^4+q^2+1+ q^{-2} + q^{-4} + q^{-6} - q^{-22} - q^{-24} - q^{-26} + q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} +2 q^{-44} +2 q^{-46} + q^{-48} -2 q^{-50} -2 q^{-52} - q^{-54} + q^{-58} + q^{-60} + q^{-62} -2 q^{-64} -2 q^{-66} - q^{-68} + q^{-72} +2 q^{-74} +2 q^{-76} - q^{-82} - q^{-84} - q^{-86} + q^{-90} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^8+q^6-1+ q^{-6} + q^{-8} }[/math]
1,1	[math]\displaystyle{ q^{20}+2 q^{16}-2 q^{10}-2 q^8+4 q^2+2+4 q^{-2} -2 q^{-8} -2 q^{-10} +2 q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{20}+q^{18}+q^{16}-q^{14}-q^{12}-q^{10}-q^8+q^4+2 q^2+2+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-18} + q^{-20} }[/math]
3,0	[math]\displaystyle{ q^{36}+q^{34}+q^{32}-2 q^{28}-2 q^{26}-2 q^{24}+q^{18}+2 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8-q^4-2 q^2-2-2 q^{-2} - q^{-4} + q^{-8} +2 q^{-10} +3 q^{-12} +3 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-24} -2 q^{-26} -2 q^{-28} + q^{-32} + q^{-34} + q^{-36} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{16}+q^{12}+q^{10}+ q^{-10} + q^{-12} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{11}+q^9+q^7-q- q^{-1} + q^{-7} + q^{-9} + q^{-11} }[/math]
1,0,1	[math]\displaystyle{ q^{26}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}-2 q^{14}-2 q^{12}-4 q^{10}-4 q^8+2 q^4+6 q^2+7+6 q^{-2} +2 q^{-4} -4 q^{-8} -4 q^{-10} -2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} +2 q^{-20} +2 q^{-22} + q^{-26} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^{10}-q^8+q^2+2+ q^{-2} - q^{-8} - q^{-10} + q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} }[/math]
1,0,0,0	[math]\displaystyle{ q^{14}+q^{12}+q^{10}+q^8-q^2-1- q^{-2} + q^{-8} + q^{-10} + q^{-12} + q^{-14} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{16}+q^{12}+q^{10}-2+ q^{-10} + q^{-12} + q^{-16} }[/math]
1,0	[math]\displaystyle{ q^{26}+q^{18}+1+ q^{-18} + q^{-26} }[/math]

B3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ q^{38}+q^{30}+q^{26}+q^{22}-q^2+1- q^{-2} + q^{-22} + q^{-26} + q^{-30} + q^{-38} }[/math]

B4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{50}+q^{42}+q^{38}+q^{34}+q^{30}+q^{26}-q^6-q^2+1- q^{-2} - q^{-6} + q^{-26} + q^{-30} + q^{-34} + q^{-38} + q^{-42} + q^{-50} }[/math]

C3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^2-2- q^{-2} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} }[/math]

C4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{28}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}-q^4-q^2-2- q^{-2} - q^{-4} + q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} + q^{-24} + q^{-28} }[/math]

D4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{38}+q^{34}+3 q^{32}+2 q^{30}+q^{28}+4 q^{26}+q^{24}-2 q^{18}-5 q^{16}-3 q^{14}-3 q^{12}-4 q^{10}+3 q^6+5 q^4+6 q^2+8+6 q^{-2} +5 q^{-4} +3 q^{-6} -4 q^{-10} -3 q^{-12} -3 q^{-14} -5 q^{-16} -2 q^{-18} + q^{-24} +4 q^{-26} + q^{-28} +2 q^{-30} +3 q^{-32} + q^{-34} + q^{-38} }[/math]
1,0,0,0	[math]\displaystyle{ q^{22}+q^{18}+q^{16}+q^{14}+q^{12}-q^2- q^{-2} + q^{-12} + q^{-14} + q^{-16} + q^{-18} + q^{-22} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{38}+q^{34}-q^{30}+q^{28}+q^{26}+q^{24}+q^{18}+q^{16}-q^{10}-q^4-1- q^{-4} - q^{-10} + q^{-16} + q^{-18} + q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} + q^{-38} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["4 1"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t- t^{-1} +3 }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-z^2 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^2+ q^{-2} -q- q^{-1} +1 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ a^2+ a^{-2} -z^2-1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^2 z^2+z^2 a^{-2} -a^2- a^{-2} +a z^3+z^3 a^{-1} -a z-z a^{-1} +2 z^2-1 }[/math]

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ -4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ -\frac{1231}{30} }[/math] [math]\displaystyle{ \frac{142}{15} }[/math] [math]\displaystyle{ -\frac{1742}{45} }[/math] [math]\displaystyle{ \frac{79}{18} }[/math] [math]\displaystyle{ -\frac{271}{30} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 4 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-2	-1	0	1	2	χ
5					1	1
3						0
1			1	1		0
-1		1	1			0
-3						0
-5	1					1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[4, 1]]
Out[2]=	4
In[3]:=	PD[Knot[4, 1]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 6, 1, 5], X[6, 3, 7, 4], X[2, 7, 3, 8]]
In[4]:=	GaussCode[Knot[4, 1]]
Out[4]=	GaussCode[1, -4, 3, -1, 2, -3, 4, -2]
In[5]:=	BR[Knot[4, 1]]
Out[5]=	BR[3, {-1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[4, 1]][t]
Out[6]=	1 3 - - - t t
In[7]:=	Conway[Knot[4, 1]][z]
Out[7]=	2 1 - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[4, 1]}
In[9]:=	{KnotDet[Knot[4, 1]], KnotSignature[Knot[4, 1]]}
Out[9]=	{5, 0}
In[10]:=	J=Jones[Knot[4, 1]][q]
Out[10]=	-2 1 2 1 + q - - - q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[4, 1], Knot[11, NonAlternating, 19]}
In[12]:=	A2Invariant[Knot[4, 1]][q]
Out[12]=	-8 -6 6 8 -1 + q + q + q + q
In[13]:=	Kauffman[Knot[4, 1]][a, z]
Out[13]=	2 3 -2 2 z 2 z 2 2 z 3 -1 - a - a - - - a z + 2 z + -- + a z + -- + a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[4, 1]], Vassiliev[3][Knot[4, 1]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[4, 1]][q, t]
Out[15]=	1 1 1 5 2 - + q + ----- + --- + q t + q t q 5 2 q t q t